matrices/script.rst
author Amit Sethi
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.. Objectives
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.. ----------
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.. At the end of this tutorial, you will be able to 
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.. 1. Create matrices using data.
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.. #. Create matrices from lists.
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.. #. Basic matrix operations.
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.. #. Use ``inv()`` function to find inverse of a matrix.
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.. #. Use ``det()`` function to find determinant of a matrix.
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.. #. Use ``eig()`` and ``eigvals()`` functions to find eigen values
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      and vectors
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.. #. Use ``norm()`` function to find norm of a matrix.
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.. #. Use ``svd()`` function to find singular value decomposition of a
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      matrix.
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.. Prerequisites
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.. -------------
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..   1. should have ``ipython`` and ``pylab`` installed. 
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..   #. getting started with ``ipython``.
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..   #. getting started with lists.
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..   #. getting started with arrays.
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..   #. accessing part of arrays.
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.. Author              : Anoop Jacob Thomas <anoop@fossee.in>
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   Internal Reviewer   : 
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   External Reviewer   :
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   Checklist OK?       : <put date stamp here, if OK> [2010-10-05]
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.. #[punch: please mark the exercises, using the syntax we decided upon.]
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========
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Matrices
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========
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{{{ show the welcome slide }}}
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Welcome to the spoken tutorial on Matrices.
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{{{ switch to next slide, outline slide }}}
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In this tutorial we will learn about matrices, creating matrices using
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direct data, by converting a list and matrix operations. Finding
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inverse of a matrix, determinant of a matrix, eigen values and eigen
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vectors of a matrix, norm and singular value decomposition of
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matrices.
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{{{ creating a matrix }}}
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All matrix operations are done using arrays. Thus all the operations
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on arrays are valid on matrices also. A matrix may be created as,
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::
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    m1 = array([1,2,3,4])
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.. #[Puneeth: don't use ``matrix``. Use ``array``. The whole script will
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.. have to be fixed.]
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Using the tuple ``m1.shape`` we can find out the shape or size of the
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matrix,
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    m1.shape
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Since it is a one row four column matrix it returned a tuple, one by
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four.
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A list can be converted to a matrix as follows,
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    l1 = [[1,2,3,4],[5,6,7,8]]
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    m2 = array(l1)
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{{{ switch to next slide, exercise 1}}}
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Pause here and create a two dimensional matrix m3 of order 2 by 4 with
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elements 5, 6, 7, 8, 9, 10, 11, 12.
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{{{ switch to next slide, solution }}}
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m3 can be created as,
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    m3 = array([[5,6,7,8],[9,10,11,12]])
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{{{ switch to next slide, matrix operations }}}
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We can do matrix addition and subtraction as,
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    m3 + m2
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does element by element addition, thus matrix addition.
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Similarly,
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    m3 - m2
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it does matrix subtraction, that is element by element
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subtraction. Now let us try,
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{{{ Switch to next slide, Matrix multiplication }}}
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::
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    m3 * m2
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Note that in arrays ``m3 * m2`` does element wise multiplication and not
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matrix multiplication,
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And matrix multiplication in matrices are done using the function ``dot()``
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    dot(m3, m2)
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but due to size mismatch the multiplication could not be done and it
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returned an error,
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{{{ switch to next slide, Matrix multiplication (cont'd) }}}
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Now let us see an example for matrix multiplication. For doing matrix
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multiplication we need to have two matrices of the order n by m and m
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by r and the resulting matrix will be of the order n by r. Thus let us
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first create two matrices which are compatible for multiplication.
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    m1.shape
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matrix m1 is of the shape one by four, let us create another one of
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the order four by two,
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    m4 = array([[1,2],[3,4],[5,6],[7,8]])
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    dot(m1, m4)
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thus the function ``dot()`` can be used for matrix multiplication.
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{{{ switch to next slide, recall from arrays }}}
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307
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As we already saw in arrays, the functions ``identity()`` which
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creates an identity matrix of the order n by n, ``zeros()`` which
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creates a matrix of the order m by n with all zeros, ``zeros_like()``
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which creates a matrix with zeros with the shape of the matrix passed,
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``ones()`` which creates a matrix of order m by n with all ones,
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``ones_like()`` which creates a matrix with ones with the shape of the
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matrix passed. These functions can also be used with matrices.
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{{{ switch to next slide, more matrix operations }}}
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To find out the transpose of a matrix we can do,
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::
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    print m4
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    m4.T
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Matrix name dot capital T will give the transpose of a matrix
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{{{ switch to next slide, Frobenius norm of inverse of matrix }}}
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.. #[punch: arange has not been introduced.]
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Now let us try to find out the Frobenius norm of inverse of a 4 by 4
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matrix, the matrix being,
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::
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    m5 = arange(1,17).reshape(4,4)
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    print m5
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The inverse of a matrix A, A raise to minus one is also called the
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reciprocal matrix such that A multiplied by A inverse will give 1. The
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Frobenius norm of a matrix is defined as square root of sum of squares
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of elements in the matrix. Pause here and try to solve the problem
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yourself, the inverse of a matrix can be found using the function
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``inv(A)``.
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And here is the solution, first let us find the inverse of matrix m5.
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::
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    im5 = inv(m5)
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.. #[punch: we don't need to show this way of calculating the norm, do
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.. we? even if we do, we should show it in the "array style".
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.. something like:
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.. sqrt(sum(each * each))]
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And the Frobenius norm of the matrix ``im5`` can be found out as,
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::
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    sum = 0
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    for each in im5.flatten():
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        sum += each * each
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    print sqrt(sum)
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{{{ switch to next slide, infinity norm }}}
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.. #[punch: similarly for this section.]
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Now try to find out the infinity norm of the matrix im5. The infinity
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norm of a matrix is defined as the maximum value of sum of the
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absolute of elements in each row. Pause here and try to solve the
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problem yourself.
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404
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The solution for the problem is,
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::
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    sum_rows = []
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    for i in im5:
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        sum_rows.append(abs(i).sum())
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    print max(sum_rows)
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{{{ switch to slide the ``norm()`` method }}}
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Well! to find the Frobenius norm and Infinity norm we have an even easier
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method, and let us see that now.
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The norm of a matrix can be found out using the method
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``norm()``. Inorder to find out the Frobenius norm of the matrix im5,
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we do,
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::
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    norm(im5)
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And to find out the Infinity norm of the matrix im5, we do,
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::
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    norm(im5,ord=inf)
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This is easier when compared to the code we wrote. Do ``norm``
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question mark to read up more about ord and the possible type of norms
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the norm function produces.
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{{{ switch to next slide, determinant }}}
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Now let us find out the determinant of a the matrix m5. 
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The determinant of a square matrix can be obtained using the function
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``det()`` and the determinant of m5 can be found out as,
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::
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    det(m5)
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{{{ switch to next slide, eigen vectors and eigen values }}}
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The eigen values and eigen vector of a square matrix can be computed
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using the function ``eig()`` and ``eigvals()``.
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Let us find out the eigen values and eigen vectors of the matrix
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m5. We can do it as,
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::
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    eig(m5)
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404
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.. #[punch: has the tuple word been introduced?]
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261
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Note that it returned a tuple of two matrices. The first element in
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the tuple are the eigen values and the second element in the tuple are
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the eigen vectors. Thus the eigen values are,
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::
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    eig(m5)[0]
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and the eigen vectors are,
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::
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    eig(m5)[1]
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The eigen values can also be computed using the function ``eigvals()`` as,
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::
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    eigvals(m5)
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{{{ switch to next slide, singular value decomposition }}}
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Now let us learn how to do the singular value decomposition or S V D
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of a matrix.
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Suppose M is an m×n matrix whose entries come from the field K, which
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is either the field of real numbers or the field of complex
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numbers. Then there exists a factorization of the form
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    M = U\Sigma V star
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where U is an (m by m) unitary matrix over K, the matrix \Sigma is an
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(m by n) diagonal matrix with nonnegative real numbers on the
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diagonal, and V*, an (n by n) unitary matrix over K, denotes the
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conjugate transpose of V. Such a factorization is called the
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singular-value decomposition of M.
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The SVD of matrix m5 can be found as
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::
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    svd(m5)
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Notice that it returned a tuple of 3 elements. The first one U the
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next one Sigma and the third one V star.
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{{{ switch to next slide, recap slide }}}
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So this brings us to the end of this tutorial. In this tutorial, we
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learned about matrices, creating matrices, matrix operations, inverse
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of matrices, determinant, norm, eigen values and vectors and singular
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value decomposition of matrices.
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{{{ switch to next slide, thank you }}}
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Thank you!
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.. 
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   Local Variables:
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   mode: rst
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   indent-tabs-mode: nil
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   sentence-end-double-space: nil
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   fill-column: 70
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   End: